About this I think you can reference to http://docs.oasis-open.org/security/saml/Post2.0/sstc-saml-tech-overview-2.0.html.
Here're my understandings about this,
with the Identity Federation Use Case to give a details for those concepts:
IdP provides the Persistent identifiers, they are used for linking to the local accounts in SPs, but they identify as the user profile for the specific service each alone. For example, the persistent identifiers are kind of like : johnForAir, jonhForCar, johnForHotel, they all just for one specified service, since it need to link to its local identity in the service.
Transient identifiers are what IdP tell the SP that the users in the session have been granted to access the resource on SP, but the identities of users do not offer to SP actually. For example, The assertion just like “Anonymity(Idp doesn’t tell SP who he is) has the permission to access /resource on SP”. SP got it and let browser to access it, but still don’t know Anonymity' real name.
The explanation for it in the spec is "The interpretation of the content of the element is left to individual implementations". Which means IdP defines the real format for it, and it assumes that SP knows how to parse the format data respond from IdP. For example, IdP gives a format data "UserName=XXXXX Country=US", SP get the assertion, and can parse it and extract the UserName is "XXXXX".